I conjecture that for two coprime integers $a$ and $b$, for any integer $n$ coprime to $a$ and $b$ they exist integers $x$ and $y$ such that
- $ax + by = n$
- $a,b,x,y,n$ are pairwise coprime
I am having problems to prove it, specially with the second condition, although numerical experimentation seems to support it. Bezout's identity doesn't guarantee it, and the only cases I have been able to identify that for sure holds for any $a,b,x,y,n$ are $x=\pm 1$ and $y=\pm 1$ for $n=\pm (a\pm b)$. I believe it could be a good starting point for some kind of inductive argument, but I have not been able to make it work in a way that guarantees the second condition.
Any help or hint to prove this conjecture, or some counterexample to it, would be welcomed.
Thanks!
We can assume without loss of generality that $\ n>0\ $, since if $\ a,b,x,y,n\ $ satisfy all the given conditions, then so do $\ a,$$\,b,$$\,{-}x,$$\,{-}y,$$\,{-}n\ $.
Using the extended Euclidean algorithm we can find integers $\ x_0,y_0\ $ such that $$ x_0a+ y_0b=1\ . $$ The integer $\ x_0\ $ must be relatively prime to $\ b\ $, and $\ y_0\ $ must be relatively prime to $\ a\ $. From the Chinese remainder theorem, it follows that there exists an integer $\ t\ $ such that \begin{align} t&\equiv -b^{-1}\pmod{n}\ \ , \ \ \ \ \text{ and}\\ t&\equiv b^{-1}(nx_0-1)\pmod{a}\ . \end{align} Let \begin{align} x=&nx_0-tb - kabn\ , \ \ \text{ and}\\ y=&ny_0+at+ka^2n\ \end{align} where $\ k $ is an integer which remains to be chosen. But whatever the value of $\ k\ $, $$ xa+yb=n\ , $$ $\ x\ $ is relatively prime to $\ b\ $, and $\ y\ $ is relatively prime to $\ a\ $, \begin{align} x&\equiv1\pmod{n}\ ,\text{ and}\\ x&\equiv1\pmod{a}\ ,\\ y&\equiv ny_0+at\equiv -b^{-1}a \pmod{n}\ . \end{align}and hence $\ x\ $ is relatively prime to $\ a\ $ and $\ n\ $ and $\ y\ $ is relatively prime to $\ n\ $. Since $\ a\ $ and $\ n\ $ are relatively prime to $\ b\ $ there is an integer $\ k\ $ such that $$ y=y_0+at+ka^2n\equiv1\pmod{b}\ . $$ Therefore, if we choose this $\ k\ $ in the equations for $\ x\ $ and $\ y\ $ above, then $\ y\ $ will be relatively prime to $\ b\ $ and the integers $\ a,b,x,y,n\ $ will satisfy all the given conditions.